If α and β are positive acute angles, sin(4α - β) = 1 and cos(2α + β) = 1/2, then the value of (α + 2β) is?

Question: If α and β are positive acute angles, sin(4α - β) = 1 and cos(2α + β) = 1/2, then the value of (α + 2β) is?

Solution:
Given,
sin(4α - β) = 1
⇒ sin(4α - β) = sin 90°
⇒ 4α - β = 90° ..... (1)

again, cos(2α + β) = 1/2
⇒ cos(2α + β) = cos 60°
⇒ 2α + β = 60° .... (2)

(1) + (2) ⇒ 4α - β + 2α + β = 90° + 60°
⇒ 6α = 150°
⇒ α = 25°
From (2), β = 60° - (2 × 25°) = 10°
∴ (α + 2β) = 25° + (2 × 10°) = 45°